Bi-Psyche-Skull

[Title thanks to the Animaniacs.]

I'm a fan of the blog Futility Closet, and a while ago their email bonus posts I subscribe to included this interesting physics problem:

I was especially curious about the angle dependency (see 4:37), so I decided to work out the mechanics.

To avoid dealing with chains and gears, let's just look at the spool shown at the same point in the video. The two forces that make it roll are the cord we're pulling, and the friction with the table:

The friction force is proportional to the normal force from the table, which is the weight of the spool, minus any vertical component of the cord force:

This frictional force applies a torque that rolls the spool to the right (clockwise), while the torque from the cord rotates the spool in the opposite direction. After a bit of algebra, the condition for rolling to the right is

Since the angles we're interested in only go between 0° and 90° (always up and right), the sine of those angles only go from 0 to 1. That means we can put limits on the pulling force to be able to choose an angle that rolls one direction or the other. The forces that allow rolling to the right are

and the forces for rolling to the left have

Some implications of this are that making the inner spool tiny always allows rolling right, but since by definition R > r, adjusting the radii can't remove the limits on rolling left. I'm a little surprised the same angle can roll left or right depending on the force. This may have to do with the fact that I have not considered whether the spool slides on the table, as well as rolling.

I scoured my apartment for something I could use to figure this out experimentally, but I couldn't find anything close enough to a spool of wire. I encourage anyone reading to try it at home and post results in the comments!